On the nonlinear Volterra equation with conformable derivative

Abstract

In this paper, we are interested to study a nonlinear Volterra equation with conformable derivative. This kind of such equation has various applications, for example physics, mechanical engineering, heat conduction theory. First, we show that our problem have a mild soltution which exists locally in time. Then we prove that the convergence of the mild solution when the parameter tends to zero.

Keywords

• conformable differential equation
• memory term
• local existence
• Banach fixed point theorem

Kaynakça

1. [1] T. Abdeljawad, On conformable fractional calculus J. Comput. Appl. Math. 279 (2015), 5766
2. [2] K. Balachandran, J.J. Trujillo, The nonlocal Cauchy problem for nonlinear fractional integrodifferential equations in Banach spaces Nonlinear Anal. 72 (2010), no. 12, 45874593
3. [3] R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative J. Comput. Appl. Math. 264 (2014), 6570
4. [4] A.A. Abdelhakim, J.A. Tenreiro Machado, A critical analysis of the conformable derivative Nonlinear Dynamics, 2019, Volume 95, Issue 4, pp 30633073.
5. [5] A. Jaiswal, D. Bahuguna, Semilinear Conformable Fractional Differential Equations in Banach Spaces Dier. Equ. Dyn. Syst. 27 (2019), no. 1-3, 313325
6. [6] M. Conti, M.E. Marchini, A remark on nonclassical diffusion equations with memory Appl. Math. Optim. 73 (2016), no. 1, 121
7. [7] X. Wang, C. Zhong, Attractors for the non-autonomous nonclassical diffusion equations with fading memory Nonlinear Anal. 71 (2009), no. 11, 57335746.
8. [8] E.C. Aifantis, On the problem of diffusion in solids, Acta Mech. 37 (1980) 265296 [9] T.W. Ting, Certain non-steady flows of second-order fluids Arch. Rational Mech. Anal., 1963, 14: 126 [10] D. Baleanu, M. Jleli, S. Kumar, B. Samet, A fractional derivative with two singular kernels and application to a heat conduction problem Adv. Difference Equ. 2020, Paper No. 252, 19 pp [11] M. Hajipour, A. Jajarmi, A. Malek, D. Baleanu, Positivity-preserving sixth-order implicit nite difference weighted essentially non-oscillatory scheme for the nonlinear heat equation Appl. Math. Comput. 325 (2018), 146158

9. [12] E. Karapinar, H.D. Binh, N.H. Luc, N.H. Can, On the continuity of the fractional derivative of the time- fractional semilinear pseudo-parabolic systems Adv. Difference Equ. 2021, Paper No. 70, 24 pp.
10. [13] R. S. Adiguzel, U. Aksoy, E. Karapinar, I.M. Erhan, On The Solutions Of Fractional Differential Equations Via Geraghty Type Hybrid Contractions, Appl. Comput. Math., V.20, N.2, 2021,313-333 [14] N.H. Tuan, N.V. Tien, D. O'regan, N.H. Can, V.T. Nguyen, New results on continuity by order of derivative for conformable parabolic equations, FRACTALS, to appear, https://doi.org/10.1142/S0218348X23400145. [15] N.H. Tuan, N.V. Tien, C. Yang, On an initial boundary value problem for fractional pseudo-parabolic equation with conformable derivative, Math. Biosci. Eng. 19 (2022), no. 11, 1123211259 [16] N.H. Tuan, T.B. Ngoc, D. Baleanu, D. O'Regan, On well-posedness of the sub-diffusion equation with a conformable derivative model, Communications in Nonlinear Science and Numerical Simulation, 89 (2020), 105332, 26 pp.
11. [17] N.A. Tuan, Z. Hammouch, E. Karapinar, N.H. Tuan, On a nonlocal problem for a Caputo time-fractional pseudoparabolic equation Math. Methods Appl. Sci. 44 (2021), no. 18, 1479114806.
12. [18] N.A. Triet, N.A. Tuan, An iterative method for inverse source parabolic equation Lett. Nonlinear Anal. Appl. Volume 1, Issue 2, Pages:7281, Year: 2023
13. [19] N.A. Triet, N.H. Tuan, Global existence for nonlinear bi-parabolic equation under global Lipschitz condition Lett. Nonlinear Anal. Appl. Volume 1, Issue 3, Pages: 89-95, Year: 2023
14. [20] M.L. Heard, S. M.RankinIII, A semilinear parabolic Volterra integro-dierential equation J. Dierential Equations 71 (1988), no. 2, 201233.
15. [21] J.V. C. Sousa, F. G. Rodrigues, E.C. Oliveira, Stability of the fractional Volterra integral-differential equation by means of ψ-Hilfer operator Math. Methods Appl. Sci. 42 (2019), no. 9, 30333043.
16. [22] H.T.K. Van, Non-classical heat equation with singular memory term, Thermal Science, Volume 25, Special issue 2, 2021